googologywikiaorg-20200223-history
User blog:Bubby3/Notation provably equivalent to BMS pair sequences
On this wiki, there are several functions that behave similarily to pair sequence system, which people don't doubt the termination of, such as the original R function, and dollar function extended bracket notation. However, people seriously doubt the termination of pair sequences of BM2.3 (or BM4), which act very similarily to them, or at least think that proving the termination is must harder than some people (including me) think. However, I think that BMS can be reformulated in terms of another notation that I will make that will be proven eqlivlant to it. Here is my notation trying to reformalize pair sequences. The notation has a number (n) and a group of nested brackets (#) with an order or subscript. It is notated n:#. 0th order brackets don't have to have a subcript. Square brackets or [] are shorthand 1st order brackets and braces {} are shorthand for 2nd order brackets. The rules are: #Replace n with n^2 #The active bracket of defined by the following process: ##If the last bracket of the expression is empty, that is the active bracket ##Othserside, repeat this process and jump into the last bracket #Let A0 be the active bracket and A(d+1) be the bracket containing Ad, until we get to an orphan bracket (a bracket not contained within another bracket) #The cases for solving the expression are: ##If the active bracket is an orphan bracket, delete ##Otherwise, if the active bracket has an order of 0, delete it and copy the parent of the former active bracket n times ##Otherwise, find the lowest m such that Am has an order of the lower than the active bracket. ###If no m exists, delete the active bracket ###Otherwise, replace the active bracket with i_n, where i_0 is empty and i_(d+1) is A(m) with the active bracket replaced with i_(d) #If the expression is empty, output the value of n, otherwise keep repeating Proof of equlivance to BMS pair sequence The definition of pair sequence system is here. We encode the expression in an expression by the following rules, done for every columm, we put a bracket with order of the bottom row, and add it as the new rightmost child of the rightmost columm whose top row is smaller than the top row of the current columm. If none such exists, we put it at the end of the expression. The tree is the way section headigs is managed on many wikis (including this one). For example, (0,0)(1,1)(2,2)(3,2)(2,1), we add all of them to make ([ []]). The active bracket coresponds to the rightmost columm. Rule 4.2 coresponds to when the bottom columm is 0, so the bad root is the parent, so the ascending matrix is all zeroes, so that means we make the copies are the same. When the last colum bottom entry is greater than zero, the bad root is indeed the outermost bracket contained in the active bracket, and the ascending value is the difference, so that means the replacement does work and the nesting does occur, Category:Blog posts